I’m not sure I understand... What would break on your local network if a public domain you own and use only for internal resources is registered by someone else

I’m not sure I understand... What would break on your local network if a public domain you own and use only for internal resources is registered by someone else? How is this different from making up a domain name? In both cases you have to set up something to resolve the names to IP addresses on your local network, be it a hosts file or DNS. I would expect that to keep working regardless of the ownership of the domain name.